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Development of Algorithmic Constructions

15:07:30 19.Apr 2024

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Polynom = x^2-7

1. Algorithm

a:=1;
b:=0;
c:=-7;


liste_max:=100000;

sieving:=proc (stelle, p)
begin
   while (stelle<=liste_max) do
     erg:=liste[stelle];
     while(erg mod p=0) do
        // Divison of the stored f(x) by the prime
        erg:=erg /p;
     end_while;
     liste[stelle]:=erg;
     stelle:=stelle+p;
   end_while;
end_proc;

// Calculation of the values of the polynom for x from 0 to liste_max
for x from 0 to liste_max do
   p:=abs (a*x^2+b*x+c);
   while (p mod 2=0) p:=p/2;
   liste [x]:=p;
end_for;

for x from 0 to liste_max do
    p:=liste[x];

    if (p>1) then

      // Printing the Primes
      print (x, p);

      // 1. Sieving
      sieving (x+p,  p);

      t:=(-x-b/a) mod p;
      if t=0 then t:=p; end_if;

      // 2. Sieving
      sieving (t, p);

    end_if;

end_for;

 2. Mathematical background
Lemma: If p | f(x) then also p | f(x+p) and  p | f(-x-b/a)

a)  p | f(x)   <=> ax^2 + bx + c = 0                       mod p
    p | f(x+p) <=> a(x+p)^2 + b(x+p) + c = 0               mod p
               <=> ax^2 + 2axp + ap^2 + bx + bp + c = 0    mod p
               <=> ax^2 + bx + c = 0                       mod p

    Thus if p | f(x) then p | f(x+p)

b)  if b = 0 mod a
    p | f(x)       <=> ax^2 + bx + c = 0                         mod p
    p | f(-x-b/a)  <=> a(-x-b/a)^2 + b(-x-b/a) + c = 0           mod p
                   <=> ax^2 + 2bx + b^2/a - bx - b^2/a + c = 0   mod p
                   <=> ax^2 + bx + c = 0                         mod p

    Thus if p | f(x)  then p | f(-x-b/a)



 3. Correctness of the algorithm
The proof for this polynom is similar to the proof for the polynom f(x)=x^2-4x+1.

a) First terms for the polynom f(x) = x^2-7

f(0)=7
f(1)=3
f(2)=1
f(3)=1
f(4)=1
f(5)=1
f(6)=29
f(7)=1
f(8)=19
f(9)=37
f(10)=31
f(11)=1
f(12)=137
f(13)=1
f(14)=1
f(15)=109
f(16)=83
f(17)=47
f(18)=317
f(19)=59
f(20)=131
f(21)=1
f(22)=53
f(23)=1
f(24)=569
f(25)=103
f(26)=223
f(27)=1
f(28)=1
f(29)=139
f(30)=1
f(31)=1
f(32)=113
f(33)=541
f(34)=383
f(35)=1
f(36)=1289
f(37)=227
f(38)=479
f(39)=757
f(40)=1
f(41)=1
f(42)=251
f(43)=307
f(44)=643
f(45)=1009
f(46)=1
f(47)=367
f(48)=2297
f(49)=1
f(50)=277
f(51)=1297
f(52)=1
f(53)=467
f(54)=2909
f(55)=503
f(56)=149
f(57)=1621
f(58)=373
f(59)=193
f(60)=3593
f(61)=619
f(62)=1279
f(63)=283
f(64)=1
f(65)=1
f(66)=4349
f(67)=1
f(68)=1
f(69)=2377
f(70)=233
f(71)=839
f(72)=167
f(73)=887
f(74)=1823
f(75)=1
f(76)=641
f(77)=1
f(78)=1
f(79)=1039
f(80)=2131
f(81)=1
f(82)=2239
f(83)=1
f(84)=1
f(85)=401
f(86)=821
f(87)=199
f(88)=2579
f(89)=1319
f(90)=8093
f(91)=197
f(92)=2819
f(93)=1
f(94)=1
f(95)=1
f(96)=9209
f(97)=1567
f(98)=457
f(99)=1

b) Substitution of the polynom
The polynom f(x)=x^2-7  could be written as f(y)= y^2-7 with x=y+0

c) Backsubstitution

Beside by backsubstitution you get an estimation for the huge of the primes with p | f(x) and p < f(x)

f'(y)>(2y-1) with with y=x+0
f'(x)>2x-1  with x > 3

4. Infinity of the sequence

5. Sequence of the polynom with 1

6. Sequence of the polynom (only primes)

7. Distribution of the primes

8. Check for existing Integer Sequences by OEIS

Found in Database : 7, 3, 1, 1, 1, 1, 29, 1, 19, 37, 31, 1, 137, 1, 1, 109, 83, 47, 317, 59,
Found in Database : 7, 3, 29, 19, 37, 31, 137, 109, 83, 47, 317, 59, 131, 53, 569, 103, 223, 139, 113, 541, 383, 1289, 227, 479, 757,
Found in Database : 3, 7, 19, 29, 31, 37, 47, 53, 59, 83, 103, 109, 113, 131, 137, 139, 149,